Question: Solve the exponential equation for $x$. 81 x + 5 3 7 x + 2 = 3 x − 2 \dfrac{81\^{ x+5}}{3\^{ 7x+2}}=3\^{ x-2} $x=$
The strategy Let's write $81$ in base $3$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $3$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 81 x + 5 3 7 x + 2 = ( 3 4 ) x + 5 3 7 x + 2 = 3 4 x + 20 3 7 x + 2 = 3 4 x + 20 − ( 7 x + 2 ) = 3 − 3 x + 18 ( 81 = 3 4 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{81\^{ x+5}}{3\^{ 7x+2}}&=\dfrac{(3^4)\^{ x+5}}{3\^{ 7x+2}}&&&&(81=3^4) \\\\\\\\ &=\dfrac{3\^{ C{4x+20}}}{3\^{ {7x+2}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=3\^{ C{4x+20} \ - \ ({7x+2})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=3\^{ -3x+18} \end{aligned} Solving the equation We obtain the following equation. 3 − 3 x + 18 = 3 x − 2 3\^{ -3x+18}=3\^{ x-2} Now we can equate the exponents and solve for $x$. $\begin{aligned} -3x+18 &=x-2\\\\ x &=5\end{aligned}$ The answer The answer is $x=5$. You can check this answer by substituting $\it{x=5}$ in the original equation and evaluating both sides.